Angular Momentum And Early Time Gluon Fields Initial state 2017 CRacow, september 20 2017 Angular Momentum And Early Time Gluon Fields Rainer J Fries Texas A&M University RJF, G. Chen, S. Somanathan, arXiv:1705:10779 S. Rose, RJF, in progress

Angular Momentum in Nuclear Collisions Initial angular momentum present at finite impact parameter 𝐿~ 𝑏 2 𝐸 𝐶𝑀 . Carried by shear movement. Mostly neglected for decades. Justified? At LHC and top RHIC energies: collision energy macroscopic, 𝐸~0.1 mJ 𝐿~ 10 6 ℏ ~ 10 −25 mJs; far from macroscopic! Initial State 2017 - Rainer J Fries x z

Angular Momentum in Nuclear Collisions Manifestations in the final state Rapidity-odd directed flow 𝑣 1 Polarization of final state particles Effects decrease with collision energy. Initial State 2017 - Rainer J Fries [STAR, Nature 548 (2017) ]

Angular Momentum in Nuclear Collisions Much recent theoretical work on vorticity, production of polarized particles, focus on lower energies. Here: how is initial angular momentum transported to midrapidity in the limit of very high energy? Use the classical McLerran-Venugopalan model as an effective theory applicable at high energies and early times Liang and Wang; Csernai, Magas and Wang; Becattini, Csernai and Wang; Pang et al.; Jiang, Lin and Liao; … Initial State 2017 - Rainer J Fries

Collisions at High Energy: Time Evolution Initial nuclear wave functions Strong classical gluon fields Local equilibration/hydrodynamization QGP/HG fluid close to local equilibrium Hadron gas and freeze-out Initial State 2017 - Rainer J Fries

Expectations Caveat: we use boost symmetry in the calculation, good approximation at high energies. Infinite system: no global conservation laws. Boost symmetry apparently does not allow for rotation, only shear flow (as in the initial state). We are used to think of flow as a vector. The situation is more interesting when rank-2 tensors are involved. Initial State 2017 - Rainer J Fries

Solving Yang-Mills Equations Color charges in nuclei → Nuclear fields 𝐴 1 𝑖 , 𝐴 2 𝑖 Initial interaction of fields (LC gauge): Here: use recursive solution for the gauge field after the collision Use high orders for realistic time evolution. McLerran and Venugoplan (1994) Kovner, McLerran, Weigert (1995) Initial State 2017 - Rainer J Fries RJF, Kapusta, Li (2006) Chen, RJF, Kapusta, Li (2015)

Energy Density and Energy Flow First: analytic estimates for event averages, lowest orders in time. Energy momentum tensor in Milne coordinates Event averages: Initial energy density Initial rapidity-even and -odd transverse flow Initial State 2017 - Rainer J Fries RJF, Kapusta, Li (2006); Chen, RJF (2013); Chen, RJF, Kapusta, Li (2015)

Energy Density and Energy Flow First: analytic estimates for event averages, lowest orders in time. Energy momentum tensor in Milne coordinates: Some components of the gluon Poynting vector carry angular momentum: They all emerge from the rapidity-odd flow term 𝛽 𝑖 . Initial State 2017 - Rainer J Fries

Angular Momentum Plot angular momentum carrying part of the Poynting vector (other terms omitted): Vortex aligned with global angular momentum Energy shear flow counter to global ang. Mom. Angular momentum: In a boost invariant system only 𝑑𝐿/𝑑𝜂 makes sense. Lowest order in time: Assuming a symmetric A+A collision. Initial State 2017 - Rainer J Fries Pb+Pb, b=6 fm Inbound longitudinal flow of L2! RJF, G. Chen, S. Somanathan, arXiv:1705:10779

Angular Momentum Back of the envelope estimate (𝜏=1/ 𝑄 𝑠 ) 𝑑 𝐿 2 𝑑𝜂 = 1 2 𝑄 𝑠 −3 𝑅 𝐴 𝜀 0 [ 𝑅 𝐴 = nuclear radius, 𝜀 0 =average energy density within radius 𝑅 𝐴 Scaling with impact parameter Qualitatively similar to other models Initial State 2017 - Rainer J Fries

Switching to Fluid Dynamics Rapid matching between classical YM and viscous fluid dynamics 𝑇 𝑓 𝜇𝜈 = 𝜀+𝑝+Π 𝑢 𝜇 𝑢 𝜈 − 𝑝+Π 𝑔 𝜇𝜈 + 𝜋 𝜇𝜈 at a time 𝜏= 𝜏 0 , similar to IP-Glasma+MUSIC Enforce conservation laws, 𝜕 𝜇 𝑇 𝑡𝑜𝑡𝑎𝑙 𝜇𝜈 =0 Keep viscous stress (or violate angular momentum conservation). Velocity field and dissipative flow contributing to angular momentum: There are other flow contributions (radial flow, Bjorken flow, etc.) that do not contribute to angular momentum (not shown). Longitudinal flow smaller than average radial flow. Initial State 2017 - Rainer J Fries

Evolution in Fluid Dynamics Further time evolution in fluid dynamics: Shear stress tensor decays Shear flow dissipates The naively expected picture holds at very late times in the collision. Boost invariance suppresses rotation and Kelvin-Helmholtz instabilities. Matching guarantees continuity of the energy momentum tensor but not of the equations of motion. Non-trivial dynamics in early times gluon fields from Gauss’ Law, switching instantaneously to a dissipation law. Initial State 2017 - Rainer J Fries VIRAL fluid dynamics 𝜂 𝑠 = 1 4𝜋 , 𝜏 0 =0.1 fm/c

Full Simulations Semi-analytic calculation Numerical sampling of the nuclear charge distributions 𝜌 1 , 𝜌 2 . Compute nuclear fields (LC gauge) 𝐴 1 𝑖 , 𝐴 2 𝑖 . Use recursion relation for the fields after the collision Time evolution of the energy density, single event Initial State 2017 - Rainer J Fries Previous numerical solutions: Krasnitz, Nara, Venugopalan (2003); Lappi (2003); … Fukushima, Gelis (2012); Schenke, Tribedy, Venugopalan (2012) Preliminary Preliminary

Angular Momentum with Fluctuations Dominated by fluctuations, dig out contributions to angular momentum Example: initial energy density, 200 events, same average charge distribution (sampled from Au+Au @ 6 fm) Directed flow can be identified, magnitude and sign consistent with the event-averaged calculation. Initial State 2017 - Rainer J Fries Relation between integrated (per rapidity) longitudinal and rapidity- odd directed flow holds. Preliminary Average 𝑇 13 𝑇 00 ~ −2% 𝑐 at 𝜏~0.3 fm → rapidity-odd directed flow of the local peak.

Conclusion and Outlook Boost invariant setup allows for angular momentum carried by gluon fields. Classical Yang-Mills phase: angular momentum built up through rapidity- odd (directed) flow of energy, driven by Gauss’ Law. Matching to viscous fluid dynamics: shear flow at midrapdity aligned with total angular momentum, dissipates in boost-invariant viscous hydro. Possible signals only from early times? Sensitivity to switch off time of classical glue dynamics. Consistent with small signals at high energies. Going to lower energies: breaking of boost-invariance needed as a next step. First results from event-by-event simulations: large fluctuations in the event plane, event-averages match up. Initial State 2017 - Rainer J Fries

Initial State 2017 - Rainer J Fries Backup

Origin of Flow in YM Initial longitudinal fields E0, B0 → transverse fields through QCD versions of Ampere’s, Faraday’s and Gauss’ Law. Here abelian version for simplicity. Gauss’ Law at fixed time t Difference in long. flux → transverse flux rapidity-odd and radial Ampere/Faraday as function of t: Decreasing long. flux → transverse field rapidity-even and curling field Full classical QCD at 𝑂 𝜏 1 : Initial State 2017 - Rainer J Fries Chen, RJF (2013)

Transverse Flow Transverse Poynting vector: Transverse plane Transverse Poynting vector: Event Plane (long. component suppressed) Radial and elliptic flow Rapidity-odd directed flow (from Gauss Law) Angular momentum Pb+Pb, b=6 fm Chen, RJF (2013) Initial State 2017 - Rainer J Fries α only β only Linear comb.

Initializing Fluid Dynamics Incomplete matching destroys energy and momentum conservation (thin lines = shear stress discarded at matching) Transverse flow dominates for matching at early times. Initial State 2017 - Rainer J Fries

Resummation of the Time Evolution Generic arguments: convergence radius of the recursive solution ~1/Qs. “Weak field” approximation to Yang Mills Can be rederived from the recursive solution. Resumming 𝑄𝜏 𝑘 terms: semi-closed form Initial State 2017 - Rainer J Fries Li, Kapusta, 1602:09060

VIRAL 3+1 D Fluid Code KT for fluxes, 5th order WENO for spatial derivatives, 3rd order TVD Runge Kutta for time integration. Bulk and shear stress, vorticity Gubser and Sod-type tests: Initial State 2017 - Rainer J Fries